The duality mapping associated to a gauge function is defined by (21).
The gauge figure is defined in three dimensions, which makes it easy to rotate.
The duality mapping J φ : E → E ∗ associated with a gauge function φ is defined by J φ ( x ) = { f ∗ ∈ E ∗ : 〈 x, f ∗ 〉 = ∥ x ∥ φ ( ∥ x ∥ ), ∥ f ∗ ∥ = φ ( ∥ x ∥ ) }, ∀ x ∈ E, where 〈 ⋅, ⋅ 〉 denotes the generalized duality pairing.
The duality mapping J φ : X ⟶ 2 X ∗ associated with a gauge function φ is defined by J φ ( x ) = { f ∗ ∈ X ∗ : 〈 x, f ∗ 〉 = ∥ x ∥ φ ( ∥ x ∥ ), ∥ f ∗ ∥ = φ ( ∥ x ∥ ), ∀ x ∈ X }, where 〈 ⋅, ⋅ 〉 denotes the generalized duality paring.
Let E ∗ be the dual space of E. The duality mapping J φ : E → 2 E ∗ associated to a gauge function φ is defined by J φ ( x ) = { f ∗ ∈ E ∗ : 〈 x, f ∗ 〉 = ∥ x ∥ φ ( ∥ x ∥ ), ∥ f ∗ ∥ = φ ( ∥ x ∥ ) }, ∀ x ∈ E. In particular, the duality mapping with the gauge function φ ( t ) = t, denoted by J, is referred to as the normalized duality mapping.
Let E* be the dual space of E. The duality mapping J φ : E → 2 E * associated with a gauge function φ is defined by J φ ( x ) = { f * ∈ E * : 〈 x, f * 〉 = ∥ x ∥ φ ( ∥ x ∥ ), ∥ f * ∥ = φ ( ∥ x ∥ ) }, ∀ x ∈ E. In particular, the duality mapping with the gauge function φ(t) = t, denoted by J, is referred to as the normalized duality mapping.
The duality mapping J φ : E → E ∗ associated with a gauge function φ is defined by J φ ( x ) = { x ∗ ∈ E ∗ : 〈 x, x ∗ 〉 = ∥ x ∥ φ ( ∥ x ∥ ), ∥ x ∗ ∥ = φ ( ∥ x ∥ ) }, ∀ x ∈ E. In the case that φ ( t ) = t, J φ = J, where J is the normalized duality mapping.
The duality mapping (J_{varphi}:Erightarrow E^) associated with a gauge function φ is defined by J_{varphi}(x)=bigl{ f^in E^:bigllangle x,f^bigrrangle =Vert x Vert varphibigl Vert xVert bigr), biglVert f^bigrVert = varphibigl Vert xVert bigr bigr}, quad forall xin E, where (langlecdot,cdotrangle) denotes the generalized duality pairing.
Gauge factor (GF) of piezoresistor is defined and simplified as: GF = frac{{frac{Delta R}{ R}}{varepsilon }{varepsilon_{eff} sigma }}{{frac{sigma }{E}}} = frac{{pi_{eff{44} E (5)where ε, π eff, π 44 and E denote the sigman, effective piezoresistive coefficient, shear piezoresistive coefficient (85 × 10−11 Pa−1) and Young's modulus of silicon (163 GPa), respectively [12].
